Besides experimental evaluations of the algorithms proposed in this work, it is also interesting to further study theoretical running time improvements. Recall that our subexponential-time algorithms for the inversions distance and the inversion-window distance are closely related to an algorithm for FAST [1]}. Feige [2]} and Karpinski and Schudy [3]} improved this algorithm for FAST by proposing algorithms with running time \(2^{\mathcal {O}(\sqrt{k})}\cdot \text{\normalfont {poly}}(n)\)  instead of \(2^{\mathcal {O}(\sqrt{k} \log (k))}\cdot \text{\normalfont {poly}}(n)\) . Therefore, a natural open question is whether \(d\) -Local W-BNSL for \(d \in \lbrace  \text{Inv}, \text{InvWin}\rbrace \)  can be solved in \(2^{\mathcal {O}(\sqrt{r})} \cdot \text{\normalfont {poly}}(|I|)\)  time.
